Fourth power
{{Short description|Result of multiplying four instances of a number together}}
{{other uses}}
In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So:
:n4 = n × n × n × n
Fourth powers are also formed by multiplying a number by its cube. Furthermore, they are squares of squares.
Some people refer to n4 as n tesseracted, hypercubed, zenzizenzic, biquadrate or supercubed instead of “to the power of 4”.
The sequence of fourth powers of integers, known as biquadrates or tesseractic numbers, is:
:0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, ... {{OEIS|id=A000583}}.
Properties
The last digit of a fourth power in decimal can only be 0, 1, 5, or 6.
In hexadecimal the last nonzero digit of a fourth power is always 1.An odd fourth power is the square of an odd square number. All odd squares are congruent to 1 modulo 8, and (8n+1)2 = 64n2 + 16n + 1 = 16(4n2 + 1) + 1, meaning that all fourth powers are congruent to 1 modulo 16. Even fourth powers (excluding zero) are equal to (2kn)4 = 16kn4 for some positive integer k and odd integer n, meaning that an even fourth power can be represented as an odd fourth power multiplied by a power of 16.
Every positive integer can be expressed as the sum of at most 19 fourth powers; every integer larger than 13792 can be expressed as the sum of at most 16 fourth powers (see Waring's problem).
Fermat knew that a fourth power cannot be the sum of two other fourth powers (the n = 4 case of Fermat's Last Theorem; see Fermat's right triangle theorem). Euler conjectured that a fourth power cannot be written as the sum of three fourth powers, but 200 years later, in 1986, this was disproven by Elkies with:
: {{math|1=206156734 = 187967604 + 153656394 + 26824404}}.
Elkies showed that there are infinitely many other counterexamples for exponent four, some of which are:Quoted in
{{cite web
| last = Meyrignac
| first = Jean-Charles
| url = http://euler.free.fr/records.htm
| title = Computing Minimal Equal Sums Of Like Powers: Best Known Solutions
| date = 14 February 2001
| access-date = 17 July 2017
}}
:{{math|1=28130014 = 27676244 + 13904004 + 6738654}} (Allan MacLeod)
:{{math|1= 87074814 = 83322084 + 55078804 + 17055754}} (D.J. Bernstein)
:{{math|1=121974574 = 112890404 + 82825434 + 58700004}} (D.J. Bernstein)
:{{math|1=160030174 = 141737204 + 125522004 + 44790314}} (D.J. Bernstein)
:{{math|1=164305134 = 162810094 + 70286004 + 36428404}} (D.J. Bernstein)
:{{math|1= 4224814 = 4145604 + 2175194 + 958004}} (Roger Frye, 1988)
:{{math|1= 6385232494 = 6306626244 + 2751562404 + 2190764654}} (Allan MacLeod, 1998)
Equations containing a fourth power
Fourth-degree equations, which contain a fourth degree (but no higher) polynomial are, by the Abel–Ruffini theorem, the highest degree equations having a general solution using radicals.
See also
References
- {{MathWorld|title=Biquadratic Number|urlname=BiquadraticNumber}}
{{Figurate numbers}}
{{Classes of natural numbers}}